Well on the moon maybe since the time lag is only one second but on Mars with a time lag of four minutes+ making on time maneuver impossible so a task batch will be sent and the robot will follow through accordingly with some flexibility for the computer to follow through autonomously.
For example robot is given instructions to build a house with step by step instructions using nuts and bolts. The robot can not lose any pieces in completing the instructions but lets say he dropped one. At this point the AI sub-routine kicks in and starts to search the missing piece then the next routine kicks in for the robot to navigate to the found piece and then another routine kicks in to retrieve the piece and lastly another routine kicks in to return to the place of origin.
There will be other autonomous instructions such as recharging batteries and or even the most mundane routine like standing back up after falling down.

I wonder which country will be seriously considering this new location for a base in or nearby?

Back in 2009, the Japanese Space Agency JAXA announced moon hole deep enough to contain a small human base. Now, the Indian Space Research Organization has discovered a "giant underground chamber" near the Moon's equator, in the Oceanus Procellarum area.

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The huge cave— discovered by the Chandrayaan-1 spacecraft—is more than one mile long (1.7 kilometers) and 393 feet wide (120 meters). By comparison, the vertical hole that Jaxa discovered was only 213 feet (65 meters across) and 289 feet deep (88 meters).

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Obviously both locations must be explored with robots and LIDAR before any construction begins.
LiDAR like this.

LiDAR technology offers one of the most accurate, expedient and cost-effective ways to capture elevation information. LiDAR data is directly processed to produce detailed bare earth models in a variety of formats

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LiDAR point clouds can be used to derive a wide variety of geospatial deliverables including:

They could generate a 3-D model of the interior cave in a 1/10th scale with 3-D printers.
With a rover with LIDAR & digital cameras they could have this 3-D data and color photos of the entire interior of this cave.

Downward-looking LIDAR instruments fitted to aircraft and satellites are used for surveying and mapping – a recent example being the NASA Experimental Advanced Research Lidar.[3] In addition LIDAR has been identified by NASA as a key technology for enabling autonomous precision safe landing of future robotic and crewed lunar landing vehicles.

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From Wikipedia.
This is how robots would land on the moon anyway.
The LIDAR could be put onto rover vehicles for closeup mapping of insid a cave chamber.

Other benefits of using a large cave:

future human base. The settlement would be protected from radiation, micro-meteor impacts, dust and extreme temperature changes by the lava structure: